Nature - USA (2020-01-23)

Nature | Vol 577 | 23 January 2020 | 489

which can be mitigated to a large extent by using nuclear notch filter-
ing^29 , leading to T 2  = 800 μs. This source of dephasing can be avoided
altogether by using group IV materials with nuclear spin-free isotopes^30.
This has led to T 2  = 28 ms for electrons in isotopically purified silicon^13 ,
and isotopic purification may also increase the quantum coherence in
germanium. Furthermore, we observe spin lifetimes of T1,Q1 = 9 μs and
T1,Q2 = 3 μs. We have found that these lifetimes increase exponentially
when lowering the tunnel coupling between each qubit and its respec-
tive reservoir (Extended Data Fig. 8), and relaxation times of T 1  > 100 μs

have been reported for germanium nanowires^19 ,^20 , both giving good

prospects for increasing the relaxation time by closing the reservoir

barrier during operation.

When the manipulation of both qubits is combined, the coupling of

the two qubits (exchange interaction J) becomes apparent. As is illus-

trated in Fig. 3a, the resonance frequency of each of the qubits is shifted

when the other qubit is prepared in its ↑ state. The strength of this

interaction depends on the inter-dot tunnel coupling t 12 as well as the

detuning ε of the dot potentials. By changing the amplitude of voltage

VH (mV)

Detuning, H

Energy

J on

J off

0

500

1,000

T^2

• (ns)

2468

2.36

2.38

2.40

f (GHz)^1

2.64

2.66

2.68

2.7

f (GHz)^3

(2,0)S (0,2)S

–1.0–0.500.5

H (meV)

H (meV)

2.40

2.42

2.44

2.46

2.48

ΔISD (fA)

2.68

2.70

2.72

2.74

2.76

fres

(GHz)

fres

(GHz)

–200 0 200

–1 01

–888 –887 –886

VBC (mV)

40

50

60

30

40

50

60

70

J/

h (MHz)

J/

N h (MHz)

ππππ

ππ

f 1 fres f 1 fres

fres fres

N

N

π

f 3 f 3 N

π

a

b

ce

d f

g

h

f 1 f 3

f 4 f 2

Fig. 3 | Tunable exchange coupling and operation at the charge symmetry
point. a, Illustration of the relevant energy levels in our hole double quantum
dot with zero (green) and finite (black) exchange coupling J between the dots.
Six energy levels are considered: the four different (1,1)-charge states as well as
the (2, 0)S and (0, 2)S singlet charge states in which both holes occupy the same
quantum dot. Four individual transitions can be driven, corresponding to the
conditional rotations of the two-qubit system. The size of the exchange
interaction is equal to J/h = f 2  − f 1  = f 4  − f 3. b, Measurement pulse cycles used to
map out the exchange splitting of Q1 (top) and Q2 (bottom). As a result of the
demodulation of the alternating cycles, transition f1,(3) gives a negative signal
and transition f2 ,(4) results in a positive signal. c, d, EDSR spectra of Q1 (c) and
Q2 (d) as a function of the detuning ε. The exchange splitting can be tuned to a
minimum at ε = 0 and increases closer to the (m, n)−(m + 1, n−1) and (m, n)−
(m − 1, n + 1) charge transitions. e, Exchange interaction as a function of ε as

extracted from c, d. Fitting the exchange coupling yields an interdot tunnel

coupling t 12 /h = 1.8 GHz and charging energy U = 1 .46 meV. f, The interdot tunnel

coupling can also be controlled by gate BC. Changing the potential on this gate,

while keeping ε = 0, allows good control over the exchange interaction between

the two qubits. g, Coherence time T* 2 of both qubits as a function of detuning

voltage Vε. When the slope of the resonance line is equal to zero, the qubit is

expected to be, to first order, insensitive to charge noise. Solid lines indicate

fits of the data to ()aTδδfresVε+ 0

−1

, with δδfVresε the numerical derivative of the

resonance line frequency as a function of detuning, T 0 the residual

decoherence and a a scaling factor. It can be observed that T* 2 is indeed longest

when the slope of the resonance line is closest to zero. Error bars correspond to

1 σ. h, Resonance frequency of transition f 1 and f 3 as a function of detuning

voltage.

Q1Q2Q1Q2Q1Q2

10 –1

100

101

Timescale (

μs)

02040

m

0

50

100

F = 99.3%

XY–X–YX/2Y/2

–X/2–Y/2

Interleaved gate

99

100

Gate delity

–20 0 100 200 300

–15

–10

P (a.u.)

0

100

200

ΔISD (fA)

ΔI

SD

(fA)

NN

tp

tp (ns)

VH

EDSR

Meas. cycle Ref. cycle

Crandom

Crandom

m

m

Cinterleave

Crecovery

Crecovery

a

b

c

e

f

d

M

R

T 2 * T 2 H T 1

Q1 Q2

Fig. 2 | Qubit control, gate f idelity and quantum coherence of planar
germanium qubits. a, Measurement sequence used for the Rabi driving
measurements. Measurement cycles with EDSR pulses are alternated with
reference cycles without a microwave tone, allowing an efficient background
current subtraction. Each cycle is repeated N times, such that measurement
and reference cycles alternate at a typical lock-in frequency of fmeas = 89.75 Hz.
b, Colour map of the differential bias current ΔISD as a function of microwave
pulse time tp and power P, where clear Rabi rotations on Q1 can be

observed. a.u., arbitrary units. c, Schematic illustration of the (interleaved)

randomized benchmarking sequence applied to Q1. C corresponds to a single

Clifford gate, with m being the total number of applied random Clifford gates.

d, Differential bias current as a function of m for the randomized benchmarking

sequence on Q1. The extracted control fidelity is FC = (99.3 ± 0.05)%. e, Gate

fidelities for the π and π/2 gates. Error bars correspond to 1σ. f, Spin coherence

and life times for Q1 and Q2. Error bars correspond to 1σ.